The evaluation of the circulation from numerical solutions of the momentum and energy equations is discussed for incompressible and compressible flows. It is shown how artificial damping directly influences the time ratio of change of the circulation

Krause, E.

English

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NASA Contractor Report 172452
lEASE REPORT NO. 84-46
i
NASA-CR-172452
19850001995 !
ICASE
REVIEW OF SOME VORTEX RELATIONS
r_iq_NCE
Egon Krause
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Contract No. NASI-17070
September 1984
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING
NASA Langley Research Center, Hampton, Virginia 23665
Operated by the UniversitiesSpace Research Association
N/LSA - ,-,LU?.[2':1984
National Aeronautics and LANGLEYRESEARCHCENTER
Space Administration " LIBRARY,NASA
LangleyResearch Center HA:,',P.TON-,VIRGINIA
Hampton,Virginia 23665
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EN,ER

REVIEW OF SOME VORTEX RELATIONS
Egon Krause
Aerodynamisches Institut, RWTH Aachen
and
Institute for Computer Applications in Science and Engineering
Abstract
The evaluation of the circulation from numerical solutions of the
momentum and energy equations is discussed for incompressible and compressible
flows. It is shown how artificial damping directly influences the time rate
of change of the circulation.
Research was supported by the National Aeronautics and Space Administration
under NASA Contract No. NASI-17070 while the author was in residence at ICASE,
NASA Langley Research Center, Hampton, VA 23665.
i

Introduction
Most finite-difference approximations of the Euler equations and of the
Navier-Stokes equations for large Reynolds numbers must resort to artificial
damping in order to stabilize the solution of the difference equations. Since
the discretization process itself and the introduction of the damping terms
changes the differential balance of mass, momentum, and energy, it is of
general interest to know to what extent the local flow behavior and the
overall flow properties are affected. If, for example, unsteady vortex
formation is to be predicted, the net spacing and the time steps must be
appropriately chosen, or else the numerical solution may completely suppress
the formation of vortices. Or, if solution branching occurs, the results
predicted may not correspond to the correct branch at all [I]. Another
example is the prediction of inviscid transonic flows. Severe falsification
of the total pressure may result in the computation if either the truncation
error is too large or the damping terms are not adequately modelled. In fact,
for irrotational flows the total pressure is often taken as a measure of
accuracy of the numerical solution.
For rotational flows, in particular for unsteady flows, such a measure is
not always available. It is then advantageous, to trace back the influence of
the artificial damping on the integral properties of the solution. As the
circulation is a characteristic integral property for rotational flows, its
dependence on the local flow properties is reviewed here in some detail.
First the material derivative of the circulation is discussed in conjunction
with the Thomson theorem. Thereby the overall influence of artificial damping
terms on the circulation can be demonstrated. By making use of various forms
of the energy equation, it is then shown how for compressible flows the
-2-
material derivative of the circulation is related to the entropy, heat
conduction, and dissipation.
2. The Material Derivative of the Circulation
As is well known (see, for example [2]), the material derivative of the
circulation r is obtained by considering its change during an infinitesimal
interval of time dt:
dr ,dV
- _ _-_ • d_ . (2.1)dt
C
Equation (2.1) can be written in a different form by splitting up the material
derivative of the velocity into its local and convective part and by
eliminating the latter with the vector identity
2
(V.V)V = V(_) - vx(Vx_). (2.2)
There results
m
dr _ _ av . ds- _ (VxCVxv)) • d_. (2.3)dt _"
c c
This relation is valid for incompressible and compressible flows, as is
equation (2.1). Another way of eliminating the material derivative of the
velocity in equation (2.1) is to replace it through the momentum equation
D
dv _ i Vp - 1 V-o" . (2.4)dt p p
-3-
In equation (2.4) the quantity o" represents the Stokes stress tensor
without the pressure. Equation (2.1) then yields
dr dp_ ! id-_ = - _ 0 _ (V-_').d_. (2.5)c
For incompressible flow the first line integral in equation (2.5),
representing the change of the static pressure along the closed curve c,
vanishes. Then dr/dt is solely determined by viscous forces, i.e.
dr _ v2
dt -v _ ( V)-d_. (2.6)
C
Equation (2.6) contains Thomson's theorem: if the kinematic viscosity
vanishes, the material derivative of the circulation is zero, i.e.
dr/dt = 0, and the circulation remains constant, if it is initially
constant. From equation (2.3) it follows for dr/dt = 0 that
av as (vx(V.v))-dW. (27)
C c
Equation (2.6) can serve to evaluate the time development of the circulation
in numerical solutions for incompressible flow.
3. Compressible Flows
The material derivative of the circulation can also be obtained for
compressible flow. Since the density is then not constant along the curve c,
the first integral in equation (2.5) does not vanish. The differential of the
-4-
pressure in equation (2.4) can be expressed by the differentials of enthalpy
and entropy according to the second law of thermodynamics,
! dp = dh - Tds, (3.1)P
and equation (2.5) becomes
dtdr_ _ Tds - _ (i ?'O')'ds. (3.2)
c C
The line integral over dh vanishes for the same reasons given in conjunction
with equation (2.6). For inviscid flows equation (3.2) reduces to [2]
dr
- _ Tds. (3.3)dt
c
In contrast to incompressible flow, circulation can be generated in
compressible inviscid•flow, as long as the entropy and temperature vary along
the path of integration in such a way that the line integral in equation (3.3)
does not vanish.
Equation (2.3) can also be derived from equation (3.3) by using Crocco's
theorem. It is obtained by expressing the static enthalpy h in equation
(3.1) through the total enthalpy and the kinetic energy
2
v
dh = dh0 - d _-- , (3.4)
and by using the vector identity, equation (2.2), and the momentum equation,
(equation (2.4))
-5-
1
TVs
= Vh0 + _7-.-d_ vx(Vxv)+ V-O'. (3.5)P
This relation is identical with Crocco's theorem if the term describing the
viscous forces is set equal to zero. If equation (3.5) is inserted in
equation (3.2), it is seen that equation (2.8) is recovered.
In numerical computations the material derivative of the circulation can
be evaluated either from equation (2.5), equation (3.2) or, for inviscid
flows, from equation (3.3). The influence of heat conduction and energy
dissipation can be demonstrated by differentiating equation (3.4) with respect
to time, and by inserting the energy equation into the resulting relation.
With
dh0 _ i Bp + ! ?.(_VT - a'.V) (3.6)dt p 8t p
there is then obtained
T ds _ 1 V.(IVT) 1
dt p - _ (o':V_) . (3.7)
A direct substitution of this relation into equation (3.2) is, however, not
possible.
4. Applications
Some of the relations discussed can be used for numerical evaluation. If
artificial damping is introduced, the momentum equation, (equation (2.4)), may
contain two additional terms of second and fourth order:
-6-
m
dtdV_ PlVp - _1V.o" + fl v2 v + f2 v4 --v' (4.1)
where fl and f2 designate the damping coefficients. The material
°
derivative of the circulation is then
dr dp I
d--{= - € O--- € _ (?'O')'ds (4.21C c
+ _ fl(V 2 V) • d_ + _ f2(V 4 V).d_.
c c
The last two integrals can be evaluated separately so that their contribution
to the circulation can be checked as a function of time. The damping of the
energy equation can be estimated from equation (3.7), if corresponding damping
terms are added.
5. Conclusions
The evaluation of the circulation in numerical solutions was discussed
for incompressible and compressible flows. It was shown that the material
derivative of the circulation is directly influenced through artificial
damping terms in the momentum equation, and indirectly through damping terms
in the energy equation.
-7-
References
[I] Krause, E.: Remarks on computationsof viscous flows, Comm. Pure Appl.
Math., Vol. XXXII, 1979, pp. 749-781.
[2] Oswatitsch, K.: Gas Dynamics, Academic Press, New York, p. 200, 1956.


1. ReportNo NASA CR-172452 j 2. Government Accession No. 3. Reclpient's Ctt_log No.
ICASE Report No. 84-46 I
4. Title and Subtitle 5. Report Date
Review of Some Vortex Relations September1984
6. Performing Organization Code
7. Author(s) 8. Performing Organ;zation ReportNo.
Egon Krause 84-46
10. Work Unit No.
9. PerformingOrganizationNameandAddress
Institute for Computer Applications in Science
and Engineering 11. Contractor GrantNo.
Mail Stop 132C, NASA Langley Research Center NASI-17070
Hampton_ VA 23665 13. Type of Report and Period Covered
12. SponsoringAgencyNameandAddress Contractor Report
NationalAeronauticsand Space Administration 14.SponsoringAgencyCode
Washington, D.C. 20546 505-31-83-01
15. Supplementary Notes
Langley Technical Monitor: J. C. South, Jr.
Final Report
16. Abstract
The evaluation of the circulation from numerical solutions of the momentum and
energy equations is discussed for incompressible and compressible flows. It is shown
how artificial damping directly influences the time rate of change of the
circulation.
i
17. Key Words(Suggestedby Author(s)) 18. DistributionStatement
circulation 34 - Fluid Mechanics & Heat Transfer
artificial damping 64 - Numerical Analysis
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